Chapter 11

Game Theory and the Tools of Strategic Business Analysis

Game Theory

0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern

0 Game theory allows us to analyze different social and economic situations

Games of Strategy Defined

0 Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player

0 A game is comprised of 0 Number of players0 Order to play0 Choices0 Chance 0 Information 0 Utility

3

Representing Games

0 Game tree0 Visual depiction0 Extensive form game0 Rules0 Payoffs

4

Game Types

0 Game of perfect information0 Player – knows prior choices

0All other players

0 Game of imperfect information0 Player – doesn’t know prior choices

5

Strategy

0 A player’s strategy is a plan of action for each of the other player’s possible actions

Game of perfect information

7Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located

1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

In extensive form

Strategies

0 IBM: 0 DOS or UNIX

0 Toshiba0 DOS if DOS and UNIX if UNIX0 UNIX if DOS and DOS if UNIX0 DOS if DOS and DOS if UNIX0 UNIX if DOS and UNIX if UNIX

9

Toshiba

(DOS | DOS,DOS | UNIX)

(DOS | DOS,UNIX | UNIX)

(UNIX | DOS,UNIX | UNIX)

(UNIX | DOS,DOS | UNIX)

IBMDOS 600, 200 600, 200 100, 100 100, 100

UNIX 100, 100 200, 600 200, 600 100, 100

Game of perfect informationIn normal form

Game of imperfect information

0 Assume instead Toshiba doesn’t know what IBM chooses0 The two firms move at the same time

0 Imperfect information0 Need to modify the game accordingly

Game of imperfect information

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Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node 3.

1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

In extensive form

Information set

Toshiba’s strategies:• DOS• UNIX

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Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Game of imperfect informationIn normal form

Another example: Matching Pennies

13

Player 2

Heads Tails

Player 1Heads - 1, +1 +1 - 1

Tails +1 - 1 - 1, +1

Extensive form of the game of matching pennies

14

Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.

Child 1

Child 2Child 2

Tails Heads

TailsHeads TailsHeads

- 1+1

+1- 1

+1- 1

-1+1

Equilibrium for GamesNash Equilibrium

0 Equilibrium 0 state/ outcome0 Set of strategies0 Players – don’t want to change behavior 0 Given - behavior of other players

0 Noncooperative games0 No possibility of communication or binding

commitments

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Nash Equilibria

chosen is *s when i player to payoff

i player of choicestrategy

choicesstrategy ofarray -

i

),...,(

),...,(*

**1

**1

n

*i

n

ss

s

sss

16

ii

nini

n

Ss

ssssss

sss

in all for

If

mequilibriu Nash a is -

ii

ˆ

),...,ˆ,...,(),...,,...,(

),...,(***

1***

1

**1

17

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Nash Equilibrium: Toshiba-IBMimperfect Info game

The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?

Dominant Strategy Equilibria

0 Strategy A dominates strategy B if0 A gives a higher payoff than B 0 No matter what opposing players do

0 Dominant strategy0 Best for a player0 No matter what opposing players do

0 Dominant-strategy equilibrium0 All players - dominant strategies

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Oligopoly Game

19

General Motors

High price Low price

FordHigh price 500, 500 100, 700

Low price 700, 100 300, 300

0 Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low0 If GM prices low, Ford is better of pricing low

Oligopoly Game

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General Motors

High price Low price

FordHigh price 500, 500 100, 700

Low price 700, 100 300, 300

0 Similarly for GM0 The Nash equilibrium is Price low, Price low

The Prisoners’ Dilemma0 Two people committed a crime and are being interrogated

separately.0 The are offered the following terms:

0 If both confessed, each spends 8 years in jail.0 If both remained silent, each spends 1 year in jail.0 If only one confessed, he will be set free while the other spends

20 years in jail.

Prisoners’ Dilemma

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Prisoner 2

confess silent

Prisoner 1Confess 8, 8 0, 20

Silent 20, 0 1, 1

0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess

Prisoners’ Dilemma

0 Each player has a dominant strategy0 Equilibrium is Pareto dominated

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Elimination of Dominated Strategies

0 Dominated strategy0 Strategy dominated by another strategy

0 We can solve games by eliminating dominated strategies

0 If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable

24

25

(a) Eliminating dominated strategies

Player 2

1 2 3

Player 11 2, 0 2, 4 0, 2

2 0, 6 0, 2 4, 0

(b) One step of elimination

Player 2

1 2

Player 11 2, 0 2, 4

2 0, 6 0, 2

(c ) Two steps of elimination

Player 2

1 2

Player 1 1 2, 0 2, 4

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(a) Eliminated dominated strategies

Player 2

1 2 3

Player 11 20, 0 10, 1 4, -4

2 20, 2 10, 0 2, -2

(b) Reduced game eliminating column 3 first

Player 2

1 2

Player 11 20, 0 10, 1

2 20, 2 10, 0

Games with Many Equilibria

0 Coordination game0 Players - common interest: equilibrium0 For multiple equilibria

0Preferences - differ 0 At equilibrium: players - no change

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28

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Games with Many Equilibria

The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX

Normal Form of Matching Numbers: coordination game with ten Nash equilibria

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Player 2

1 2 3 4 5 6 7 8 9 10

Player 1

1 1, 1 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

3 0, 0 0, 0 3, 3 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

4 0, 0 0, 0 0, 0 4, 4 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

5 0, 0 0, 0 0, 0 0, 0 5, 5 0, 0 0, 0 0, 0 0, 0 0, 0

6 0, 0 0, 0 0, 0 0, 0 0, 0 6, 6 0, 0 0, 0 0, 0 0, 0

7 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 7, 7 0, 0 0, 0 0, 0

8 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 8, 8 0, 0 0, 0

9 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 9, 9 0, 0

10 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 10, 10

Table 11.12

A game with no equilibria in pure strategies

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General 2

Retreat Attack

General 1 Retreat 5, 8 6, 6

Attack 8, 0 2, 3

The “I Want to Be Like Mike” Game

31

Dave

Wear red Wear blue

Michael Wear red (-1, 2) (2, -2)

Wear blue (1, -1) (-2, 1)

Credible Threats

0 An equilibrium refinement:0 Analyzing games in normal form may result in equilibria

that are less satisfactory0 These equilibria are supported by a non credible threat0 They can be eliminated by solving the game in extensive

form using backward induction0 This approach gives us an equilibrium that involve a

credible threat0 We refer to this equilibrium as a sub-game perfect Nash

equilibrium.

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33

Toshiba

(DOS | DOS,DOS | UNIX)

(DOS | DOS,UNIX | UNIX)

(UNIX | DOS,UNIX | UNIX)

(UNIX | DOS,DOS | UNIX)

IBMDOS 600, 200 600, 200 100, 100 100, 100

UNIX 100, 100 200, 600 200, 600 100, 100

Non credible threats: IBM-ToshibaIn normal form

0 Three Nash equilibria0 Some involve non credible threats.0 Example IBM playing UNIX and Toshiba playing UNIX

regardless:0 Toshiba’s threat is non credible

Backward induction

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1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

Subgame perfect Nash Equilibrium

0 Subgame perfect Nash equilibrium is0 IBM: DOS0 Toshiba: if DOS play DOS and if UNIX play UNIX

0 Toshiba’s threat is credible0 In the interest of Toshiba to execute its threat

Rotten kid game

0 The kid either goes to Aunt Sophie’s house or refuses to go

0 If the kid refuses, the parent has to decide whether to punish him or relent

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Player 2 (a parent)

(punish if the kid refuses)

(relent if the kid refuses)

Player 1(a difficult

child)

Left(go to Aunt Sophie’s House)

1, 1 1, 1

Right(refuse to go to Aunt Sophie’s House)

-1, -1 2, 0

Rotten kid game in extensive form

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• The sub game perfect Nash equilibrium is: Refuse and Relent if refuse• The other Nash equilibrium, Go and Punish if refuse, relies on a non

credible threat by the parent

Kid

Parent

RefuseGo to Aunt Sophie’s House

Relent if refuse

Punish if refuse

-1-1

20

11

1

2